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\title{探照灯镜面的微分方程}
\author{五六七}
%\date{2025年9月21日}

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\section{探照灯镜面的微分方程}
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\begin{frame}[allowframebreaks]{探照灯镜面的微分方程}


\vspace{-0.3cm}
  
我们要设计一个旋转曲面（反光镜）的形状，使得从点光源发出的光线经其反射后，变成平行于某一轴的光束（如探照灯所需的平行光）。


\begin{figure}[h]
    \centering
    \begin{tikzpicture}[scale=1.0]
        \begin{axis}[
            axis lines = middle, % 设置坐标轴位置
            xlabel = $x$, ylabel = $y$, % 设置x轴和y轴标签
            ymin=-10, ymax=10, % 设置y轴范围
            xmin=-10, xmax=10, % 设置x轴范围
            domain=-1:2, % 函数定义域
            samples=40, % 样本点数量，值越大曲线越平滑
            %grid=major, % 显示网格线
            width=6cm, height=6cm, % 设置图像大小
            %enlargelimits=true, % 放大坐标轴一点以确保所有的数据点都能显示
            >={Stealth[scale=1.4]}, % 设置箭头样式
            ]
            
            % 绘制函数 y^2=2(2-x)
            \addplot [thick, blue] {4*sqrt(2-x)};
            \addplot [thick, blue] {-4*sqrt(2-x)};
            
            % 如果需要添加更多函数或数据点，可以继续添加 \addplot 命令
            % 例如：\addplot [red, mark=*] coordinates {(0,0) (1,1) (2,4)};

            \draw[->,>={Stealth[scale=1.2]}, red] (0,0) -- (1,4);
            \draw[->,>={Stealth[scale=1.2]}, red] (1,4) -- (-1,4);
            
        \end{axis}
    \end{tikzpicture}
%    \caption{反射镜面}
%    \label{fig:mirror-reflection}
\end{figure}


\newpage 

问题重述

点光源位于某固定点（通常设在原点或焦点处）；

反光镜是绕 $x$-轴（或光轴）旋转而成的曲面；

要求：所有从光源发出、经镜面反射的光线都平行于 $x$-轴；

求：该旋转曲面的母线（即截面曲线）$ y = y(x) $ 的形状。

\newpage 

几何设定

我们建立二维平面模型（三维为绕轴旋转）：

设点光源位于坐标原点 $ O(0, 0) $；

反光镜的母线为一条平面曲线 $ y = y(x) $，定义在 $ x \leq c $ 上；

任取曲线上一点 $ P = (x, y) $；

光线从 $ O(0,0) $ 射向 $ P(x,y) $，入射方向为 $ \vec{OP} = (x, y) $；

经镜面反射后，要求反射光线水平向左，即方向为 $ (-1, 0) $. 

目标：根据反射定律，推导出曲线 $ y = y(x) $ 所满足的微分方程，并求解其形状。

\newpage 

\begin{figure}[h]
    \centering
    \begin{tikzpicture}[scale=0.7]

        % 绘制 x 轴 y 轴
        \draw[->,>={Stealth[scale=1.2]}] (-2,0) -- (6,0) node[right] {$x$};
        \draw[->,>={Stealth[scale=1.2]}] (0,-2) -- (0,6) node[above] {$y$};

        % 绘制向量
        \draw[->,>={Stealth[scale=1.2]},red] (0,0) -- (1,2);% node[above] {$(x,y)$};
        \draw[->,>={Stealth[scale=1.2]},blue] (1,2) -- (2,4) node[above] {$(x,y)$};

        \draw[->,>={Stealth[scale=1.2]},red] (1,2) -- (-1.235,2);% node[above] {$(x,y)$};
        \draw[->,>={Stealth[scale=1.2]},blue] (1,2) -- (3.235,2) node[right] {$(1,0)\cdot\sqrt{x^2+y^2}$};
        
        \draw[->,>={Stealth[scale=1.2]},blue] (1,2) --++ (3.235,2) node[right] {$k(dy,-dx)$};

        % 绘制切线
        \draw[dashed,purple] (1,2) -- ++ (2,-3.235);
        \draw[dashed,purple] (1,2) -- ++ (-2,3.235);

        % 标记点和标签
        \node at (0,0) [below left] {O};
        \draw[purple] (3,-1.5) node [right] {{镜面切线}};
        \draw[red] (-1.2,2) node [left] {{反射光线}};

    \end{tikzpicture}
%    \caption{}
%    \label{fig}
\end{figure}


\newpage 

得到母线曲线所满足的一阶常微分方程
$$
\frac{x + \sqrt{x^2 + y^2} }{y} = \frac{dy}{-dx}.
$$

求解可得 
$
y^2 = 4p(c-x), x\le c.
$

反光镜应为以点光源为焦点的旋转抛物面，其方程为 
$$
y^2 + z^2 = 4p(c-x).
$$

\newpage 

探照灯、汽车大灯：灯泡放在抛物面反射镜的焦点处，发出的光经反射后形成平行光束，照射远处。

卫星天线、望远镜：反过来使用——平行入射信号（如电磁波）被抛物面聚焦到焦点处的接收器。


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